Spectral and Product Growth in Graphs and Groups

Definitions and mathematical statement

The source poses three related growth questions.

1. In the usual finite undirected graph setting with at least two vertices, let $M$ be the adjacency matrix and order its eigenvalues algebraically as $\lambda_1\geq\lambda_2\geq\cdots$. Obtain lower bounds for the adjacency spectral gap $\lambda_1-\lambda_2$. The original question explicitly asks for lower bounds; no graph class, degree, size parameter, normalization, or proposed bound is specified.
2. For a finite subset $A$ of an abelian group written additively, define \[ A+A=\{a+b:a,b\in A\}. \] How large is $|A+A|$ compared with $|A|$, and what bounds or structural explanations are available?
3. For a finite subset $A$ of an arbitrary group written multiplicatively, define \[ AAA=\{abc:a,b,c\in A\}. \] How large is $|AAA|$ compared with $|A|$, and why?

For the two set-growth questions the nonempty case is the substantive one; no assumption that $A$ generates its ambient group, is symmetric, or avoids proper subgroups is supplied. These are broad requests, not a single specified expansion theorem. In particular, disconnected graphs and finite subgroups are not excluded by the literal wording, so a uniform positive gap or a uniform strictly superlinear growth conclusion would require additional hypotheses. The record does not select such hypotheses, a target exponent, or an exceptional structured class.

Short English statement

Seek lower bounds for adjacency spectral gaps and understand sumset and triple-product growth relative to set size. The graph and group hypotheses needed for a nontrivial general bound are not specified.

Sources

• Local UnsolvedMath registry, permanent ID $30000951$, source code catalogue entry, "Spectral and Product Growth in Graphs and Groups." No individual proposer is supplied.
• Groups and Geometries (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/20, pp. 1031--1078, also listed as "Oberwolfach Report 20/2008": https://doi.org/10.4171/owr/2008/20.
• Supplied reference, "Helfgott, growth in SL3": https://arxiv.org/abs/0807.2027.
• Supplied reference, "Helfgott, growth in SL2": https://doi.org/10.4007/annals.2008.167.601.
• Supplied literature triage, checked 2026-08-21: status "partially solved." It credits Helfgott and others with major product-growth and spectral-gap results for important finite groups including $\operatorname{SL}_2$ and $\operatorname{SL}_3$, while describing the broad questions as open. Precise hypotheses are not supplied. This dated assessment derives from supplied OWR curation; time-sensitive status and cited claims require independent verification.